On Structures Of Semirings And Congruences On Semirings

In this dissertation, we give a definition of translational hull of additive commutative semiring; besides ,we give a characterization of constructure of strongly distributive lattice of semirigs; finally,we give a definition of kernel normal system of inverse semiring. The main results are given in follow.In Chapter 1, we give the introduction and preliminaries.In Chapter 2, we give a definition of translational hull of additional commutative semiring at first; besides we discuss the relation of translational hulls according to the relation of semirings.The main results are given in follow.Define 2.1 The left map A : S → S is called a left translation of 5 if for allx,y e S,λ(x+ y) = λx +λy,λ(xy) = (λx)y.The right map p : S → S is called a right translation of S if for all x,y ∈S, (x + y)ρ = xρ + yρ, (xy)ρ = x(yρ).Theorem 2.6 Let θ : S → T be epimorphism, kerθ be a congruence on S which keeps consistency after left translation and right translation, then σ : Ω(S) → Ω(T), (λ, ρ) → (λ', ρ') is morphism,where λ't = λ'(sθ) = (λs)θ,tρ' = (sθ)ρ' = (sρ)9,sθ = t,t ∈ T. If λ and ρ are permutable,then λ' and ρ' are per-mutable.In Chapter 3,we give a definition of strongly distributive lattice of semirings;besides,we give a characterization of constructure which is subdirect product of a distributive lattice and a ring. The main results are given in follow .Define 3.1 Let D be a distributive lattice,{S_α|α ∈ D} be a family of pairwise disjoint semirings indexed by D.For each pair α ≥ β in D,there exists a monomor-phism Ψ_α,β : Sα → S_β such that:atp1ppn = ^a,7> a > P > 75On 5 = UQ6￡)SQ,the operations are defined as follows:Va G 5Q,6 Gab = aifia a + b = c,c€ and c satisfiesThe system is called strongly distributive lattice of semirings. We write S —,define 9 on 5:a e SQ,b e S0,a6b & aipa,ap = bipptOc0.Then 0 is a congruence on 5,and 5 is subdirect product of a distributive lattice and a ring, Va G D, SQ is (left,right,weakly)cancellative,then 5/5 have the same characterization.In chapter 4,we give a definition of kernel normal system of an inverse semiring at first and we have kernel normal system and congruenee which are-one to one;besides,we characterize a congruence which seperates the additive idem-potents by kernel normal system;finally,we give kernel normal system of strongly distributive lattice of inverse semirings. The main results are given in follow.Let N— {N{\i € /} be a family of pairwise disjoint additive Inverse subsemir-ings of S indexed by I . J\f is defined to be a kernel normal system of inverse semiring 5,if:(.Rl)each idempotent of S is contained in some element of M; {R2)Va G 5,Vi G I,a + iV, + a C Njt3j € /;We denote j = i + o,so that a + Ni + a C Ni+a; (i?3)If a, a + 6, b + b' G A^then b G N{; (R4)Vx G 5, Ni eN^,xNi C Nj, NiX C Nk> 3j, he I.Theorem 4.5 Let M— {Nt\i G /} is a kernel normal system of inverse semiring S, then p^ is a congruence on S,M is a kernel of ptf .Conversely,^ : S —> T is epimorphism ,JV is additive kernel of <￡,then M is a kernel normal system of p^ and pm—^P ? V?"1-Theorem 4.15 If p is a congruence which seperates additive idempotents on inverse semiring S, then additive kernel ftf of p is additive group kernel normal system of S and p=px .Conversely,if M is additive group kernel normal system of S,thus there is a p, which satisfy that Af is additive kernel of p and p is a congruence on S which seperates additive idempotents ,p—PM ■Theorem 4.19 Let S =< D;Sa^a,fi >,{M?}qgd are permitting kernel normal system of 5,then :M={N = [JQ&DNa\Na e Ma,Ng ￡ M0,Na^a>a0 = N^0i